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Theorem isldil 34197
Description: The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
ldilset.b 𝐵 = (Base‘𝐾)
ldilset.l = (le‘𝐾)
ldilset.h 𝐻 = (LHyp‘𝐾)
ldilset.i 𝐼 = (LAut‘𝐾)
ldilset.d 𝐷 = ((LDil‘𝐾)‘𝑊)
Assertion
Ref Expression
isldil ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝐻(𝑥)   𝐼(𝑥)   (𝑥)

Proof of Theorem isldil
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ldilset.b . . . 4 𝐵 = (Base‘𝐾)
2 ldilset.l . . . 4 = (le‘𝐾)
3 ldilset.h . . . 4 𝐻 = (LHyp‘𝐾)
4 ldilset.i . . . 4 𝐼 = (LAut‘𝐾)
5 ldilset.d . . . 4 𝐷 = ((LDil‘𝐾)‘𝑊)
61, 2, 3, 4, 5ldilset 34196 . . 3 ((𝐾𝐶𝑊𝐻) → 𝐷 = {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)})
76eleq2d 2672 . 2 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)}))
8 fveq1 6086 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2611 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑥) = 𝑥 ↔ (𝐹𝑥) = 𝑥))
109imbi2d 328 . . . 4 (𝑓 = 𝐹 → ((𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1110ralbidv 2968 . . 3 (𝑓 = 𝐹 → (∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
1211elrab 3330 . 2 (𝐹 ∈ {𝑓𝐼 ∣ ∀𝑥𝐵 (𝑥 𝑊 → (𝑓𝑥) = 𝑥)} ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥)))
137, 12syl6bb 274 1 ((𝐾𝐶𝑊𝐻) → (𝐹𝐷 ↔ (𝐹𝐼 ∧ ∀𝑥𝐵 (𝑥 𝑊 → (𝐹𝑥) = 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899   class class class wbr 4577  cfv 5789  Basecbs 15643  lecple 15723  LHypclh 34071  LAutclaut 34072  LDilcldil 34187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ldil 34191
This theorem is referenced by:  ldillaut  34198  ldilval  34200  idldil  34201  ldilcnv  34202  ldilco  34203  cdleme50ldil  34637
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